Note that for any two compact subsets <math>K_1 \subseteq K_2 \subseteq \Omega,</math> the natural inclusion <math>\operatorname{In}_{K_1}^{K_2} : C^{k}\left( K_1; Y \right) \to C^{k}\left( K_2; Y \right)</math> is an embedding of TVSs and that the union of all <math>C^{k}\left( K; Y \right),</math> as ''K'' varies over the compact subsets of <math>\Omega,</math> is <math>C_c^{k}\left( \Omega; Y \right).</math>
=== Space of compactly support ''C''<sup>''k''</sup> functions ===
For any compact subset <math>K \subseteq \Omega,</math> let <math>\operatorname{In}_{K}_K : C^{k}\left(K; Y \right) \to C_c^{k}\left(\Omega; Y \right)</math> be the natural inclusion and give <math>C_c^{k}\left(\Omega; Y \right)</math> the strongest topology making all <math>\operatorname{In}_{K}_K</math> continuous.
The spaces <math>C^{k}\left(K;Y \right)</math> and maps <math>\operatorname{In}_{K_1}^{K_2}</math> form a [[direct limit|direct system]] (directed by the compact subsets of <math>\Omega</math>) whose limit in the category of TVSs is <math>C_c^{k}\left(\Omega;Y \right)</math> together with the natural injections <math>\operatorname{In}_{K}.</math>{{sfn | Trèves | 2006 | pp=412-419412–419}}
The spaces <math>C^{k}\left( \overline{\Omega_i}; Y \right)</math> and maps <math>\operatorname{In}_{\overline{\Omega_i}}^{\overline{\Omega_j}}</math> also form a [[direct limit|direct system]] (directed by the total order <math>\mathbb{N}</math>) whose limit in the category of TVSs is <math>C_c^{k}\left(\Omega;Y \right)</math> together with the natural injections <math>\operatorname{In}_{\overline{\Omega_i}}.</math>{{sfn | Trèves | 2006 | pp=412-419412–419}}
Each natural embedding <math>\operatorname{In}_{K}_K</math> is an embedding of TVSs.
A subset ''S'' of <math>C_c^{k}\left(\Omega;Y \right)</math> is a neighborhood of the origin in <math>C_c^{k}\left(\Omega;Y \right)</math> if and only if <math>S \cap C^{k}\left(K;Y \right)</math> is a neighborhood of the origin in <math>C^{k}\left( K; Y \right)</math> for every compact <math>K \subseteq \Omega.</math>
This direct limit topology on <math>C_c^{\infty}\left(\Omega \right)</math> is known as the '''canonical LF topology'''.
If ''Y'' is a Hausdorff locally convex space, ''T'' is a TVS, and <math>u : C_c^{k}\left(\Omega;Y \right) \to T</math> is a linear map, then ''u'' is continuous if and only if for all compact <math>K \subseteq \Omega,</math> the restriction of ''u'' to <math>C^{k}\left(K;Y \right)</math> is continuous.{{sfn | Trèves | 2006 | pp=412-419412–419}} One replace "all compact <math>K \subseteq \Omega</math>" with "all <math>K := \overline{\Omega_iOmega}_i</math>".
